www.ietdl.org Published in IET Microwaves, Antennas & Propagation Received on 23rd May 2012 Revised on 18th October 2012 Accepted on 7th November 2012 doi: 10.1049/iet-map.2012.0283
ISSN 1751-8725
Rigorous derivation of lossy equivalent circuit for narrowband waveguide direct-coupled-cavity filters Jose Antonio Lorente1, Christoph Ernst2, Alejandro Alvarez Melcon1 1
Departmento Tecnologias de la Informacion y las Comunicaciones, Universidad Politecnica de Cartagena, Antiguo Cuartel de Antigones 30202 Cartagena, Spain 2 TEC-ETM ESA/ESTEC Kepleerlan 1, 2202 AZ Noordwijk ZH, The Netherlands E-mail:
[email protected]
Abstract: This study presents a technique to rigorously derive a complete lossy equivalent circuit of narrowband direct-coupledcavity filters with arbitrary cross-section and arbitrary coupling geometries. Traditionally, only dissipation loss and stored energy because of closed unperturbed cavities have been taken into account in the design of complete filters and the contribution because of coupling structures has been neglected. An additional challenge is the non-trivial discrimination of cavities and couplings in a direct-coupled-cavity filter. With the proposed circuit, the contribution of the coupling structures to the total dissipated power and total stored energy of the filter can be derived and a clear separation between resonators and couplings is established. The technique developed uses the even and odd mode theory to successfully derive lossy equivalent circuits for inter-cavity couplings. The lossy circuit of external (input/output) couplings is also derived. A fifth degree direct-coupled-cavity filter has been designed, simulated and manufactured. Its lossy equivalent circuit has been extracted and comparisons with simulations and measurements show excellent agreement. Applications of the novel equivalent circuit in the prediction of losses in waveguide filters and in yield analyses are also illustrated.
1
Introduction
Today most demanding applications require very stringent loss specifications in the design of microwave filters. Owing to their low dissipation loss and high-power handling capability, microwave waveguide cavity filters are widely used in many applications. Hence, an accurate and fast loss prediction of a filter can be decisive when very tight specifications are to be satisfied. To do so, an accurate circuit model that completely characterises a directcoupled-cavity filter including losses because of resonators and coupling elements is derived in this paper. The broad number of publications presented for the derivation [1–3] or use of equivalent circuits for microwave filters [4, 5] confirms the importance of deriving an accurate network representation for this kind of structures. The applications that require the use of equivalent circuits are diverse and include the derivation of expressions for the sensitivity of coupled resonator filters [4] or the derivation of methods for the tunning of RF microwave filters [5], among others. On the other hand, all the previous publications centred their work in lossless filters. When a network including losses is required, the work in [6] can be of help. However, the approach requires m = 2n simulations of the complete nth-order filter structure to derive a Vandermonde matrix, which is often ill-conditioned and, if the number of points are increased, its resolution could be problematic. The new approach presented in this work overcomes these difficulties IET Microw. Antennas Propag., 2013, Vol. 7, Iss. 4, pp. 251–258 doi: 10.1049/iet-map.2012.0283
by a proper segmentation of the complete structure, so the analyses of only individual parts are needed to extract the equivalent lossy circuit. A classical direct-coupled-cavity filter (see an example in Fig. 1a) can be represented by the equivalent circuit in Fig. 1b for narrow pass bands [1] containing halfwavelength resonators and frequency-independent impedance inverters. Losses in the resonators are characterised by their unloaded Q factors, whereas the couplings (impedance inverters) are traditionally assumed to be lossless and frequency independent. Since for narrowband filters the cavities are weakly coupled they closely resemble unloaded resonators; hence it has been common practice since the fifties to calculate the dissipation losses of the filter from the unloaded Q of the resonators [7]. It is well known that this does not yield the absolute losses in the filter, but applying a safety factor between 50 and 80% good agreement with measurements can be achieved (note this factor includes deviations because of the effects of surface roughness). However, this type of calculations is only approximate. Clearly, the correction factors are needed since dissipation in the coupling apertures is neglected and it is also assumed that the perturbed cavity can be represented by a closed un-perturbed cavity, which might not be the best option in the pursuit of optimised filter geometries and topologies with reduced insertion loss [8]. A more rigorous approach to compute the losses in a given filter structure separating couplings and cavities and assigning 251
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Fig. 1 Classical representation of direct-coupled filters a Waveguide filter b Circuit representation with lossless couplings
the corresponding losses to each element is considered here. The losses in the cavities will be represented by the attenuation factor in the associated transmission lines, whereas lumped element circuit models will be used for the calculation of the losses in the couplings. A number of publications have been reported on the characterisation of both external and inter-resonator couplings [8–14]. Deslandes and Boone [9] proposed an equivalent lossy circuit for inter-resonator couplings, and by using an even and odd formalism they were able to characterise a structure comprised of two resonators coupled together. However, a rigorous derivation of the equivalent circuit is missing. Also, the technique used in [9] for the calculation of the external coupling, does not differentiate between stored energy and losses in cavity and coupling. Miraftab and Yu [10] have proposed an equivalent circuit of lossy couplings based on complex impedance inverters [11], but a derivation of the circuit elements is omitted. Besides, the paper is centred on advanced synthesis techniques of lossy filtering functions and the focus is not in the derivation of the lossy equivalent circuit of a filter. Other publications centred their work in the characterisation of the external coupling with the goal to measure the unloaded Q factor of a cavity. On the one hand, [12, 13] propose an equivalent circuit where losses because of the coupling mechanism are taken into account, but the resonant frequency deviation because of the coupling is not corrected. In this context, Thal [8] uses a similar circuit for the characterisation of losses in different coupling geometries. This type of circuits is used under the assumption that same coupling loadings yield to same coupling coefficients. However, this assumption is not valid for different iris thicknesses or completely different coupling geometries and the theory in [8] can no longer be applied. On the other hand, this deviation is taken into account in [14], but the coupling is assumed to be lossless. In this work the ideas in [9] are further developed and a step-by-step derivation of an equivalent circuit based on transmission line resonators is obtained by deriving the equivalent circuits of the filter couplings. A practical technique to derive the lossy equivalent circuit of inter-cavity coupling apertures is presented, and in contrast to [8, 9], the derivation of the lossy external coupling is also included in this work. This allows us to achieve excellent agreement of the extracted equivalent circuit response of a filter with its full-wave response. Also, the agreement in the rejection band is improved because of the use of transmission line resonators. Thus, the equivalent circuit can be used for the tuning of microwave lossy filters. 252 & The Institution of Engineering and Technology 2013
The different and novel formulation presented here allows separating resonator and coupling contributions to the overall losses. With this approach it is then possible to individually study the loss contributions of the resonators and coupling structures of the complete filter. This can be useful, for example to find optimised geometries of cavities and coupling structures, which might yield to insertion loss reductions of even more than 20% [8, 15]. The techniques described can be applied to any arbitrarily shaped coupling structure and waveguide cross-section. Measured and simulated results of a filter of fifth-order confirm the excellent accuracy obtained with the new circuit. The derived circuit is also applied to the loss prediction of practical waveguide coupled-cavity filters of different orders, bandwidths and centre frequencies with minimum use of full-wave simulation. Finally, an application of the new circuit to yield analysis is also presented in this work, showing a tremendous reduction in the use of lengthy full-wave simulations.
2 Step-by-step derivation of the equivalent lossy circuit The approach used to derive the complete equivalent lossy circuit of a filter consists on deriving the equivalent lossy circuits of the individual couplings and assemble them together to yield the final equivalent lossy network of the complete structure. To that end, the unloaded resonator, the inter-resonator coupling, and the external coupling must be studied separately. Each process is introduced in the next subsections. 2.1
Unloaded resonator
The first step is to characterise the isolated resonator that will be used in the final equivalent circuit. A transmission line resonator can be seen as a portion of a lossy transmission line with l = λg0/2. For rectangular resonators, the fundamental TE10 waveguide mode can be used for the derivation of the equivalent circuit in analytic form. In the more general case of arbitrary resonator cross-section, the complex propagation constant of the dominant mode (γC = αC + jβC) can be directly extracted from full-wave simulations. The rest of the parameters can then be derived numerically from the propagation constant (characteristic impedance ZC, and the reactance slope parameter χC) [11]. 2.2
Inter-resonator coupling
To derive the equivalent lossy circuit of an inter-resonator coupling, a symmetric model as the one used in [9] consisting of two cavities coupled through a waveguide iris can be used. However, an analysis of the even and odd fields of the structure reveals that a virtual magnetic wall can be placed in the electrical centre of both cavities without perturbing the electromagnetic fields. It can be shown that the use of magnetic walls allows us to perform a more rigorous characterisation of the inter-resonator couplings of a filter. In addition, by placing these magnetic walls in the electrical centre, the problem can be reduced as shown in Fig. 2a; thus saving computational time and allowing an accurate calculation of the overall losses. In contrast to previous work [9], transmission line resonators are used in the equivalent circuit. Note that the length of IET Microw. Antennas Propag., 2013, Vol. 7, Iss. 4, pp. 251–258 doi: 10.1049/iet-map.2012.0283
www.ietdl.org right and left of the AA′ plane in Fig. 2c [11]. This equation reads Qodd =
xodd + xTL.odd
